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The controversy was not so much about the tension between vis viva and mechanics, as about what is the "true" quantity of motion, vis viva or momentum, and what is the "metaphysical" basis upon which mechanics is to be built, vis viva or Newton's force (aside from multiple other side issues that got entangled with it, see What was the vis ...
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In the paper Einstein's controversy with Drude and the origin of statistical mechanics: A new glimpse from the "Love Letter" by J. Renn, (1997), we read:
The nature of Einstein's objections to Drude's theory [...] has remained just as unknown to us as the character of Drude's response to a letter we know Einstein had written to him around early ...
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Mirowski P., More heat than light (1989 Cambr.UP), Chap.2. pp.11-98 The history of the energy concept.< It's a great book for various other reasons.>
For Ernst Mach, energy was more or less defined as the ability to do
work. Although "work" could mean many things to many people, Mach felt
it was merely an historical accident that the term ...
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The notion of moment is important in rigid body dynamics. Historically speaking, this was pioneered by Archimedes in his Method where he outlined his theory of the lever.
It's worth adding that the parallelogram of forces/velocities was pioneered in the work known as Mechanica, traditionally thought to be Aristotle, but now thought to be by a student of his ...
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In my opinion: Euler did so much already that the contributions from later physicists/mathematicians are at a level of abstraction that is beyond the scope of standard physics courses.
Among the subjects not already covered by Euler, I assume, is the quite unique case of the intermediate axis theorem. Arguably awareness of that phenomenon took off only ...
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Once you have calculus and if we can assume that the tension is a smooth function of displacement, it is easy enough to discover Hookes law, it's simply the statement that to a good approximation that the tension is proportional to the displacement for small displacements. The physics comes in determining the constant of proportionality.
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The derivation from solid state physics is quite simple. At equilibrium, atoms are at distances where the first derivative of potential vanishes. So for small deviations, a series expansion is dominated by the quadratic term.
Force is the derivative of potential. Far a harmonic potential, the force increases linearly with distance from the equilibrium ...
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